Weighted random sampling and reconstruction in general multivariate trigonometric polynomial spaces

The set of sampling and reconstruction in trigonometric polynomial spaces will play an important role in signal processing. However, in many applications, the frequencies in trigonometric polynomial spaces are not all integers. In this paper, we consider the problem of weighted random sampling and reconstruction of functions in general multivariate trigonometric polynomial spaces. The sampling set is randomly selected on a bounded cube with a probability distribution. We obtain that with overwhelming probability, the sampling inequality holds and the explicit reconstruction formula succeeds for all functions in the general multivariate trigonometric polynomial spaces when the sampling size is sufficiently large.

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of $L_{p,{1 / \omega }}$ L p , 1 / ω -norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.

Sampling and Reconstruction of Continuous-Time Signals

This chapter presents the theory for transferring a continuous-time signal into its discrete-time form by sampling, and then converting the obtained samples to a digital signal suitable for processing in a processing machine, using the procedure of sample quantizing and coding. Then, the procedure of converting a digitally processed signal into discrete signal samples and the reconstruction of the initial continuous-time signal via a lowpass reconstruction filter is presented. The theory provides the mathematical base for both analogue-to-digital and digital-to-analogue conversions, which are extensively used for processing signals in discrete communication systems. The chapter goes on to show that the Nyquist criterion must be fulfilled to eliminate signal aliasing in the frequency domain. Finally, the mathematical model for transferring a continuous-time signal into its discrete-time form, and vice versa, is presented and demonstrated for a sinusoidal signal.

Average sampling in mixed shift-invariant subspaces with generators in hybrid-norm spaces

This paper mainly studies the average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue spaces $L^{p,q}(\mathbb{R}^{d+1})$, under the condition that the generator $\varphi$ of the shift-invariant subspace belongs to a hybrid-norm space of mixed form, which is weaker than the usual assumption of Wiener amalgam space and allows to control the orders $p,q$. First, the sampling stability for two kinds of average sampling functionals are established. Then, we give the corresponding iterative approximation projection algorithms with exponential convergence for recovering the time-varying shift-invariant signals from the average samples.